On the equational theory of finite modular lattices

TL;DR

This paper demonstrates the undecidability of identity validity in finite modular lattices for a fixed number of variables, linking group theory problems to lattice theory.

Contribution

It establishes the existence of a bound beyond which no algorithm can decide identity validity in finite modular lattices, using techniques from group theory.

Findings

No algorithm can decide identities in finite modular lattices beyond a certain variable limit.

The result relies on the unsolvability of the Restricted Word Problem for finite groups.

The paper connects group presentation problems with the equational theory of modular lattices.

Abstract

It is shown that there is $N$ such that there is no algorithm to decide for identities in at most $N$ variables validity in the class of finite modular lattices. This is based on Slobodskoi's result that the Restricted Word Problem is unsolvable for the class of finite groups and relies on Freese's technique of capturing group presentations within free modular lattices.